Spectral-to-Automorphic Geometric Langlands Functor

• The spectral-to-automorphic geometric Langlands functor is a categorical transform that maps spectral data from moduli stacks of Langlands parameters to automorphic sheaves using Hecke symmetries.
• It intertwines tensor operations on the spectral side with Hecke actions on the automorphic side, ensuring compatibility across global, local, and tamely ramified settings.
• The construction leverages Whittaker normalization and nilpotent singular-support conditions to establish an equivalence between spectral and automorphic categories.

Searching arXiv for the cited papers to ground the article in current literature. Using the arXiv search tool to retrieve the relevant papers. Searching for core references on the spectral-to-automorphic geometric Langlands functor. The spectral-to-automorphic geometric Langlands functor is the categorical transform that carries spectral data on a stack of Langlands parameters to automorphic sheaves on a moduli stack of bundles, while intertwining tensor operations on the spectral side with Hecke symmetries on the automorphic side. In the unramified global setting over a smooth projective curve $X$, it is the functor

$$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$

and the recent proof of the unramified geometric Langlands conjecture identifies this functor with an equivalence of categories. In the categorical local Langlands setting for quasisplit $p$-adic groups, an explicit spectral-to-automorphic functor

$$t_\psi:\operatorname{Coh}(\operatorname{Par}_G)_{\mathrm{fin}}\to D(\mathrm{Bun}_G)$$

is constructed on a decisive finite subcategory and is identified with the restriction of a right adjoint $R_\psi$ to the Langlands functor $L_\psi$ (Ben-Zvi, 22 May 2026, Hansen et al., 31 May 2026).

1. Global categorical formulation

In the global geometric Langlands correspondence, $G$ is a connected complex reductive group and $X$ is a smooth projective curve over an algebraically closed field. The automorphic stack is $\mathrm{Bun}_G(X)$, the moduli of principal $G$-bundles on $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$0, and its automorphic category is the derived DG category of sheaves on $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$1. In the de Rham form, this is $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$2, the derived category of $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$3-modules on $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$4; in the Betti form, it is a constructible sheaf category with nilpotent singular-support condition, often denoted $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$5 (Ben-Zvi, 22 May 2026).

The spectral stack is the moduli of $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$6-local systems on $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$7. In Betti form it is

$$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$8

while in de Rham form it is the stack of flat $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$9-bundles. Its natural categorical realizations include $p$0 and the ind-coherent refinement $p$1, where the nilpotent condition is part of the correct global formulation (Ben-Zvi, 22 May 2026).

In the unramified setting, the precise equivalence is stated as

$p$2

with $p$3 in the de Rham setting and the nilpotent constructible category in the Betti setting. The spectral-to-automorphic functor $p$4 is the functor realizing this equivalence from the spectral to the automorphic side (Ben-Zvi, 22 May 2026).

This formulation is a precise refinement of earlier descriptions of geometric Langlands as a categorical equivalence between coherent sheaves on a moduli stack of local systems and $p$5-modules on $p$6, characterized by Hecke/Wilson compatibility and by sending skyscraper sheaves at points of the spectral stack to Hecke eigensheaves on the automorphic side (Frenkel, 2012).

2. Characterizing properties of the functor

Conceptually, $p$7 is characterized by two principles. The first is normalization by the Whittaker object: $p$8 Here $p$9 is the spectral unit, and $$t_\psi:\operatorname{Coh}(\operatorname{Par}_G)_{\mathrm{fin}}\to D(\mathrm{Bun}_G)$$0 is the Whittaker sheaf on $$t_\psi:\operatorname{Coh}(\operatorname{Par}_G)_{\mathrm{fin}}\to D(\mathrm{Bun}_G)$$1, described as the “white light” object representing the Whittaker period (Ben-Zvi, 22 May 2026).

The second principle is intertwining of symmetries. If $$t_\psi:\operatorname{Coh}(\operatorname{Par}_G)_{\mathrm{fin}}\to D(\mathrm{Bun}_G)$$2 acts on $$t_\psi:\operatorname{Coh}(\operatorname{Par}_G)_{\mathrm{fin}}\to D(\mathrm{Bun}_G)$$3 by Hecke functors and on $$t_\psi:\operatorname{Coh}(\operatorname{Par}_G)_{\mathrm{fin}}\to D(\mathrm{Bun}_G)$$4 by tensoring with vector bundles from the universal local system, then for any $$t_\psi:\operatorname{Coh}(\operatorname{Par}_G)_{\mathrm{fin}}\to D(\mathrm{Bun}_G)$$5,

$$t_\psi:\operatorname{Coh}(\operatorname{Par}_G)_{\mathrm{fin}}\to D(\mathrm{Bun}_G)$$6

Equivalently, using the universal $$t_\psi:\operatorname{Coh}(\operatorname{Par}_G)_{\mathrm{fin}}\to D(\mathrm{Bun}_G)$$7-local system $$t_\psi:\operatorname{Coh}(\operatorname{Par}_G)_{\mathrm{fin}}\to D(\mathrm{Bun}_G)$$8,

$$t_\psi:\operatorname{Coh}(\operatorname{Par}_G)_{\mathrm{fin}}\to D(\mathrm{Bun}_G)$$9

This is the formal statement that $R_\psi$0 carries tensor operations on the spectral side to Hecke convolution on the automorphic side (Ben-Zvi, 22 May 2026).

Geometric Satake underlies this identification. For a fixed point $R_\psi$1, the Hecke stack parametrizes one-point modifications of bundles, and the Hecke functor is schematically

$R_\psi$2

These functors assemble into a symmetric monoidal action of the spherical Hecke category, and geometric Satake yields an equivalence

$R_\psi$3

so that irreducible Hecke operators correspond to tensoring by irreducible $R_\psi$4-representations (Ben-Zvi, 22 May 2026).

A parallel characterization appears in the local categorical Langlands program. There one starts with the enhanced Whittaker coefficient functor $R_\psi$5, its left adjoint $R_\psi$6, and then defines $R_\psi$7 as the unique $R_\psi$8-linear ind-completion lifting $R_\psi$9 on compact objects. The right adjoint $L_\psi$0 preserves colimits and compact objects, and on finite coherent sheaves the explicit functor

$L_\psi$1

satisfies

$L_\psi$2

This makes $L_\psi$3 the local $L_\psi$4-adic analogue of a spectral-to-automorphic transform on a controlled spectral subcategory (Hansen et al., 31 May 2026).

3. Hecke eigensheaves, Langlands parameters, and nilpotent singular support

A point $L_\psi$5 is a Langlands parameter. For any representation $L_\psi$6 of $L_\psi$7, the associated local system on $L_\psi$8 is $L_\psi$9. The Hecke eigencondition for an automorphic sheaf $G$0 with parameter $G$1 is

$G$2

for every $G$3 and $G$4, compatibly and functorially in $G$5. Factorization in $G$6 makes the family of Hecke operators commute, so the correspondence is described as a categorical spectral theorem in which Hecke operators are commuting Hamiltonians, eigensheaves are monochromatic states, and points of $G$7 are colors or frequencies (Ben-Zvi, 22 May 2026).

From this viewpoint, $G$8 implements the global diagonalization. It sends skyscraper-type spectral objects supported at $G$9 to Hecke eigensheaves with parameter $X$0, and it sends the spectral unit to the Whittaker sheaf. This is the sense in which geometric Langlands is described as a nonabelian algebraic spectral theorem (Ben-Zvi, 22 May 2026).

The nilpotent singular-support condition is essential in the precise formulation. On the spectral side, $X$1 is singular, so the correct category is not plain $X$2 but $X$3, and the correspondence requires restriction to $X$4. In the global unramified theorem this nilpotent condition is described as the functional-analytic fine-tuning ensuring that spectral and automorphic categories match (Ben-Zvi, 22 May 2026).

The same principle appears on the automorphic side in the Betti theory. For sheaves $X$5 on $X$6 with singular support inside the global nilpotent cone $X$7, Hecke modifications do not introduce cotangent directions along $X$8. More precisely,

$X$9

with zero cotangent along $\mathrm{Bun}_G(X)$0. This local constancy in the modification point yields a symmetric monoidal action

$\mathrm{Bun}_G(X)$1

which establishes the “automorphic to Galois” direction in Betti geometric Langlands (Nadler et al., 2016).

A plausible implication is that the global functor $\mathrm{Bun}_G(X)$2 and the Betti spectral action are two manifestations of the same organizing principle: the spectral category acts because Hecke operators become locally constant under nilpotent singular-support hypotheses, and the spectral-to-automorphic transform packages that action into actual eigensheaf production (Nadler et al., 2016, Ben-Zvi, 22 May 2026).

4. Constant terms, Eisenstein series, and gluing

The spectral-to-automorphic functor is constrained by parabolic functoriality. On the automorphic side, for a parabolic $\mathrm{Bun}_G(X)$3, one has geometric Eisenstein and constant term functors. In the global $\mathrm{Bun}_G(X)$4-module setting,

$\mathrm{Bun}_G(X)$5

The automorphic gluing theorem shows that $\mathrm{Bun}_G(X)$6 is reconstructed from tempered pieces attached to $\mathrm{Bun}_G(X)$7 and its standard Levi subgroups, and that $\mathrm{Bun}_G(X)$8 and $\mathrm{Bun}_G(X)$9 preserve tempered objects while $G$0 preserves anti-tempered objects (Beraldo et al., 2022).

This gluing result is designed to match the spectral gluing theorem. On the spectral side,

$G$1

while on the automorphic side the corresponding glued category is built from $G$2 and enhanced constant-term functors. The paper on automorphic gluing states that given tempered Langlands functors for Levi subgroups, spectral and automorphic gluing assemble them into the full functor $G$3, reducing the full conjecture to the tempered conjecture (Beraldo et al., 2022).

A more direct commuting property is sketched for the global de Rham spectral-to-automorphic functor $G$4. For a parabolic $G$5 with Levi $G$6, the asserted compatibility is

$G$7

where $G$8 is the pull-push integral transform along

$G$9

The proof strategy uses Hecke structures on geometric Eisenstein series, compatibility of Jacquet functors with the geometric Casselman–Shalika equivalence, and factorization-localization from $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$00 to $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$01 (Faergeman et al., 18 Jul 2025).

In the categorical local Langlands conjecture, parabolic compatibility is built into the formalism of $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$02, $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$03, and $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$04. For standard $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$05,

$$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$06

and on finite coherent sheaves

$$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$07

These identities make explicit that the spectral-to-automorphic transform is not merely Hecke-equivariant but also compatible with the parabolic architecture of the Langlands program (Hansen et al., 31 May 2026).

5. Betti, local, and tamely ramified variants

The Betti form of geometric Langlands posits a dg equivalence

$$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$08

for a compact Riemann surface $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$09 with underlying oriented topological surface $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$10. In this setting, the spectral-to-automorphic functor is expected to be a canonical dg equivalence

$$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$11

compatible with Hecke symmetries, mapping class group actions, parabolic induction, gluing along pants decompositions, and $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$12 structures arising from topological field theory (Ben-Zvi et al., 2016).

A local Betti analogue is now proved in the tame setting for the universal affine Hecke category. The main theorem identifies

$$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$13

as an equivalence of monoidal dg-categories. This is described as the tamely ramified local Betti geometric Langlands equivalence for the universal affine Hecke category, and specializes to the unipotent monodromy regime as another argument for Bezrukavnikov’s theorem (Dhillon et al., 24 Jan 2025).

In the $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$14-adic categorical local Langlands program, the spectral parameter stack is $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$15, the Artin stack of continuous $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$16-cocycles $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$17 modulo $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$18-conjugation. The automorphic side is $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$19, the DG category of lisse $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$20-sheaves on the stack of $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$21-bundles on the Fargues–Fontaine curve. The construction of $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$22 depends on admissible ind-coherent sheaves, admissible duality $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$23, the Chevalley involution twist $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$24, and the enhanced Whittaker coefficient $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$25. Its basic adjunction identity is

$$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$26

and on finite coherent sheaves

$$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$27

These formulas identify $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$28 as a canonical and explicit spectral-to-automorphic transport on a large subcategory decisive for the conjecture (Hansen et al., 31 May 2026).

A tamely ramified global de Rham variant is also developed for curves over $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$29. There the automorphic category is $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$30, built from $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$31-bundles with $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$32-reductions at marked points and character-sheaf equivariance, while the spectral stack is $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$33 for regular singular local systems with prescribed local eigenvalues. The paper proves an action

$$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$34

compatible with Hecke operators both on $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$35 and at the marked points, and proves existence of a coherent Hecke eigensheaf $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$36 for any irreducible regular-singular $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$37 (Færgeman, 2024).

6. Status, examples, and conceptual significance

The unramified global equivalence has been proved in a sequence of works by Gaitsgory–Raskin and collaborators, with the conceptual blueprint organized around factorization and geometric Satake, construction and normalization of the Langlands functor by the Whittaker period, the Kac–Moody/opers machine for cuspidal eigensheaves, Eisenstein compatibility, and the necessity of $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$38. The final outcome is described as multiplicity-one spectral decomposition: every color $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$39 appears with a unique eigensheaf, and $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$40 realizes the equivalence between spectral and automorphic categories (Ben-Zvi, 22 May 2026).

In the local $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$41-adic theory, the full categorical local Langlands conjecture is proved for $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$42 under the Eisenstein–Whittaker compatibility hypothesis, and an induction principle reduces the general case to proper Levi subgroups together with a small amount of information about $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$43. Under “very well-understood” hypotheses and the same compatibility, the induction principle applies to many quasisplit classical groups in types $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$44, $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$45, $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$46, and $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$47 (Hansen et al., 31 May 2026).

Concrete global constructions outside the full theorem also illustrate the spectral-to-automorphic direction. For $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$48, a backward functoriality construction via geometric theta-lifting produces nonzero Hecke eigensheaves on $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$49 from $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$50-local systems whose standard representation is an irreducible rank-$$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$51 local system on $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$52. The resulting eigensheaf is canonical up to a $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$53-decomposition (Lysenko, 2019). In the tamely ramified de Rham setting, the existence of Hecke eigensheaves for irreducible regular-singular local systems is used to prove motivicity of irreducible rigid $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$54-local systems with quasi-unipotent monodromies and finite order abelianization (Færgeman, 2024).

Historically, the spectral-to-automorphic transform has long been constrained by trace-formula heuristics and by Hecke/Wilson compatibility. Earlier surveys described the categorical equivalence as a transform sending $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$55 on the spectral stack to a Hecke eigensheaf $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$56, and related the behavior of the functor to geometric and relative trace formulas, Whittaker sheaves, and integral-transform kernels (Frenkel, 2012). This suggests that the functor is not only an equivalence statement but also a mechanism organizing spectral decomposition, functoriality, and categorical traces across the Langlands program.

A recurring misconception is that the functor is simply “tensor by a kernel” in the naive Fourier–Mukai sense. The literature instead presents a more rigid structure: normalization by Whittaker data, Hecke equivariance via geometric Satake, nilpotent singular-support constraints, parabolic compatibility, and duality compatibility are all part of the definition or characterization. Another misconception is that $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$57 alone suffices on the spectral side. The precise unramified theorem, the gluing formalism, and the local categorical theory all point to ind-coherent refinements and finiteness conditions as essential rather than auxiliary (Ben-Zvi, 22 May 2026, Beraldo et al., 2022, Hansen et al., 31 May 2026).

In current usage, the expression “spectral-to-automorphic geometric Langlands functor” therefore denotes a family of closely related constructions—global $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$58, local $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$59 and $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$60, tame $$\Phi:\ \operatorname{IndCoh}_{\mathrm{Nilp}}\big(\operatorname{LocSys}_{G^\vee}(X)\big)\ \longrightarrow\ \mathsf{Shv}(\mathrm{Bun}_G),$$61, and Betti TFT-based variants—whose common content is the transport of spectral parameter data into automorphic sheaf theory in a way that intertwines Hecke operators, Whittaker normalization, parabolic functoriality, and duality.